Present Value and the Opportunity Cost of Capital Fundamental Question by ramhood4

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									Present Value and the Opportunity
Cost of Capital
    Fundamental Question: How are asset prices
       •   Companies invest in a variety of real assets.
       •   Objective is to find real assets which are worth more than
           they cost.
       •   Essence of the capital budgeting process.
       •   Asset valuation's relationship to a well-functioning capital
           market: If there is a good market for an asset, its value is
           exactly the same as its market price.
    Present Value Example (B&M):
        Suppose your house burns down leaving you with a lot worth
        $50,000 and a check for $200,000.
        Option 1: Rebuild
        Option 2: Build an office building; cost is $350,000; you predict you
        can sell for $400,000 in one year.
        •    Rule: You should go ahead if the present value of the expected
             $400,000 payoff is greater than the investment of $350,000.

        Question: What's the value today of $400,000 one year from now?
                 PV = Discount Factor x C1
                    Discount Factor = 1/1+r , where r is the reward
                     investors demand for accepting delayed payment.

        Assume that the $400,000 payoff one year from now is a sure thing.

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        Now, how else could you make $400,000 one year from now?
           •    Invest in government securities (assume 7% ROR)
           •    How much must you invest?
                $400,000/1.07 = $$373,382
        •  Now assume you've committed land and begun construction -
           now you decide to sell. How much would you sell it for?
        •  Since it produces $400,000, well-informed investors would be
           willing to pay $373,832.
        •  NOTE: This is the only feasible price that satisfies buyer and
           seller. Present Value is also the market price.


    Net Present Value

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             Found by subtracting required investment from present value of
             cash flows.
                      NPV = C0 + C1/(1+r)

    Net Present Value Rule:
         Accept investments that have a positive NPV.
    Rate of Return Rule:
         Accept investments that offer rates of return in
        excess of their opportunity cost of capital.
    The NPV rule and the IRR rule can conflict when
       we have multiple period cash flows.
    Present Value Techniques
             •   Valuation of Cash Flows (components)

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                   1.   Magnitude
                   2.   Timing

              •   Timeline
              Example: We wish to calculate the PV of the cash flow stream,
                  $100 received in one year and $200 received in two years,
                  where r = 10%

                             $100                 $200

            t =0             t=1                   t=2

              PV = $100/(1+.10) + $200/(1+.10)2 = $256

        •     Discount Factors (Appendix Table 1 in B&M) - Very useful
              when the same discount or interest rate is applied over the life of
              the project.

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                            Discount              Cash             Present
         Period             Factor                Flow             Value
         ----------         ------------          ------------     -----------
         0                  1.0                   -$150,000        -$150,000
         1                  1/1.07 = .935         -$100,000        -$ 93,500
         2                  1/(1.07)2 = .873      +300,000         +261,900
                                                  Total NPV =      + $18,400

    Intuition Behind Present Value
              •       Think about the problem in terms of future value.

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             •   If you have $100 and put it in the bank for one year at 6%,
                 how much do you have after one year?
             •   So, $100 received now is equivalent to $106 received in
                 one3 year.
             •   Thus, to translate money from its value at t=0 to its value at
                 t=1, we multiply by (1+r)
             •   Correspondingly, to translate value from t=1 to t=0, we
                 divide by (1+r).
             •   This notion is easily extended to multiple time periods.
             •   Provides a powerful means to value long-lived assets.
                      Oil and gas reserves
                      Value of a mine

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Perpetuities and Annuities
             • Securities issued by the British government
             • No obligation to repay

             • Offer fixed income for each year to perpetuity.

             • Rate of return is equal to promised annual payment divided

               by PV -       Return = Cash flow/PV
             • Rearrange, and find the PV, given discount rate and annual

               cash payment
             EXAMPLE: Suppose you have $1000 in bank drawing 5%; how
               much can you withdraw at year end and still have same
               amount in bank as at the beginning of the year? This process
               can continue forever or in "perpetuity".
             EXAMPLE: Endowed chair at CSM; r = 10% and annual cash
               payment for chair is $100,000.
             PV of perpetuity = C/r = $100,000/0.10 = $1,000,000

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    Growing Perpetuity - Allows for growth in annual
       cash payment.

             PV = C1/(1+r) + C1(1+g)/(1+r)2 + C2(1+g)2/(1+r)3 + . . .

             Simple formula for this is the sum of a geometric series:
             PV of growing perpetuity = C1/(r-g)

             EXAMPLE: Endowed chair where g = 0.04.
             PV = C/(r-g) = $100,000/(0.10-0.04) = $1,666,666

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             • An asset that pays a fixed sum each year for a specific
               number of years, i.e. house mortgage.
             • An annuity that makes payments in each of years 1 to t is

               equal to the difference between two perpetuities.
             PV of Annuity (for year 1 to year t):
                    C/r - [(C/r)(1/(1+r)t)]
                    C[(1/r) - (1/(r(1+r)t)]
               NOTE: Expression in brackets is the annuity discount factor,
                 which is the PV at discount rate, r, for an annuity of $1
                 paid at the end of each of t periods.
               EXAMPLE: Endowed chair - cost to endow chair for 20
                 years at $100,000 per year?
               PV = $100,000[(1/.10) - (1/(.10(1.10)20)]
                   = $100,000 x 8.514
                   = $851,400

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    Compund vs. Simple Interest
               •   Compund Interest - each interest payment is reinvested to
                   earn more interest in subsequent payments.
               •   Simple Interest - no opportunity to earn interest on interest.

               NOTE: Discounting is a process of compound interest.
               EXAMPLE: What is the PV of $100 to be received 10 years
                 from now, if the opportunity cost of capital is 10%?
                     PV = 100/(1.10)10 = $38.55

           •   Simple interest and continuously compounded interest are rarely
               used in the business world. For virtuallly every application,
               compound interest is the accepted technique.

           Rule of 70: Number of years that it takes for an investment to
             double in value at an annually compounded interest rate of r, is
             approximately 70/r. So, if interest rate is 10%, then 7 years is
             the amount of time.

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Valuing Stocks and Bonds
    Key Issues
             •   Focus on cash flows (magnitude and timing)
             •   Caclculate the PV of those cash flows

             • Fixed set of cash payoffs - usually semi-annual.
             • Interest payments and face value at maturity.

             • Treasury bonds, municipal bonds, corporate bonds, foreign

               bonds, mortgage bonds
             EXAMPLE: Suppose in August of 1989, you invest in a 125/8%
             1994 U.S. Treasury bond.
                   Coupon rate = 125/8%
                   Face Value = $1000
                   Each year you receive .1265 * 1000 = $126.50
                   In August, 1994 you receive $1126.25
                                           CASH FLOWS

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            1990             1991               1992          1993            1994
            $126.25          $126.25            $126.25       $126.25         $1126.25
              WHAT IS THE 1989 MARKET VALUE OF THIS STREAM?Compare
      return to similar securities.Medium term Treasury bonds at that time offered ROR
      of 7.6%
              NOTE: 7.6% ROR is what investors gave up to invest in bond.
                                       5 Ct
                               PV = ∑            t = $1202.77
                                     t =1(1+ r )
              NOTE: Bond price quotes are expressed in a percentage of face value; i.e.
             $1202.77 = 120.28 percent
       Consider the alternative view: If the price of the bond is $1202.77,
              what return do investors expect?
               1202.77 = $126.25 + $126.25 + $126.25 + $126.25 + $1126.25
                            1+ r       (1+ r )2 (1+ r )3 (1+ r )4    (1+ r )5
                                So, r = 7.6%
                 • r is yield to maturity or internal rate of return

              NOTE: Bond yields are usually quoted as semi-annual compond rates.

    Valuation of Common Stock

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             •   Cash payoffs to owners of common stock
                        - cash dividends
                        - capital gains or losses
                        PO = current price
                        P1 = expected price
                        Div1 = expected divdend

             Rate of return, r, that investors expect from a share over the next
             year is defined as the expected dividend per share, Div1, plus
             the expected price appreciation per share, P1 - P0, all divided by
             the beginning year price.
                                         Div1 + P1 − P0
                              ER = r =
                   •   r is the market capitalization rate or expected return
                       on a security.

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             EXAMPLE: Company X is selling for $100/share; investors
             expect a $5 dividend and the stock to sell for $110 one year
             from now. The expected return, r, then is:

                            r = $5+ $110 − $100 = 0.15

             Correspondingly, given investors' forecasts of dividend and price and the
             expected return offered by other equally risky stocks, then today's price:
                                           Div1 + P1
                            Price = P0 =             If r, the expected return on securities
                                            1+ r
             in the same risk class as Firm X is 15 percent, then today's price should
                            P0 = $5+ $110 = $100NOTE: All securities in an
            equivalent risk class are priced to offer the same return. No other price
            could survive in competitive capital markets.
        Multiple Period Valuation

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             Just as price at P0 is dependent on investors expectations about
             year 1's dividends and stock price appreciation, so also can we
             compute P1, based on expectations about year 2.
                                     Div2 + P2
                              P1 =
                                       1+ r

             EXAMPLE: Suppose a year from now investors will be looking
             at dividends of $5.50 in year 2 and a subsequent price of $121.
             This would imply a price at the end of year 1 of:

                              P1 = $5.50 + $121 = $110

             Today's price then can be computed either as:

                              Div1 + P1 $5.00 + $110
                       P0 =            =             = $100
                               1+ r          .

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                    Div1 Div2 + P2 $5.00 $5.50 + $121
             P0 =       +         =     +             = $100We can
                    1+ r (1+ r )2    .
                                    115       .
                                            (115) 2

             extend this result to multiple periods in the following manner:
                              H     Divt         PH
                         P0 = ∑              +
                             t = 1 (1+ r )     (1+ r ) H

                    where H is the final period and the summation is the
                    discounted present value of the dividend stream from year
                    1 to H.

             NOTE: Stock valuation is nothing more than the sum of the
             present value of the expected cash flows from dividend
             streams during the period and the present value of the end-
             of-period stock price.

             In the case where H goes to infinity, we can ignore P because
             this value goes to 0.
             So, then we can think of P0 as the present value of a perpetual
             stream of cash dividends, where

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                                        ∞   Divt
                                 P0 = ∑
                                     t = 1 (1 = r )

    Growth Rate and the Gordon Model
             Dividend growth Rate
             Incorporation of growing perpetuities

                                 P0 = r − g

            g = anticipated growth rate (must be < r)
            r = market capitalization rate
            NOTE: As g approaches r, stock price becomes infinite - r
            must be greater than g if growth is perpetual.
        Estimating Capitalization Rate

        Market capitalization rate, r, equals the dividend yield (Div1/P0)
        plus the expected rate of growth in dividends (g), where

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                            r=        +g

        EXAMPLE (B&M): Sears, Roebuck & Company is selling for
        $42/share early in 1989; dividend payments for 1989 are expected to
        be $2.00 per share.

                                    Div1 $2.00
                 Dividend Yield =       =      = 0.048
                                     P0 $42.00

        What about g, the rate of growth in dividends?


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        1. Observe Historical payout ratio, ratio of dividends to earnings per
           share. for Sears, historically been about 45%; 55% of earnings is
           reinvested: plowback ratio.

        2. Sears' return on equity, ROE, or ratio of earnings per share to book
           equity per share is about 12%.
                       ROE =                      = 0.12
                               BookEquity / Share

        3. Assume fairly stable ROE and payout ratios.
        4. If we forecast 12% ROE and reinvest 55%, then book equity
           should increase by 0.55 x 0.12 = 0.066. (Since we assume that
           ROE and payout are constant, earnings and dividends per share
           will also increase by 6.6%). Therefore dividend growth rate is:
                   g = plowback ratio x ROE = 0.55 x 0.12 = 0.066

        So, utilizing the Gordon Model, we may now compute the market
          capitalization rate, r.

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                      r=         + g = 0.048 + 0.066 = 0.114
        or about 11.5% is the market capitalization rate.

        Methodological shortcomings of the Gordon Model:
             1. Constant growth?
             2. Constant Plowback?
             3. Constant ROE?
             4. Single stock analysis may be inappropriate - may need to
                look at a class of stocks, i.e. FERC sets market
                capitalization rates for utility industry.
             5. Rule of thumb only.
             6. Avoid use when firm is in a high growth period
                (Remember, g is a long-term growth rate.)

    Stock Price and Earnings per Share

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        •   Growth Stock - those stocks purchased primarily for the
            expectation of capital gains; interested in future growth or earnings
            rather than next year's dividends.
        •   Income Stock - purchased primarily for the cash dividends.

              Consider the case of a company that does not grow at all, i.e. no
              plowback of earnings - simply produces a constant stream of
              dividends. Special case where:
              Expected return = dividend yield = earnings-price ratio
                                                      Div1 EPS1
                        Expected Return =                 =
                                                       P0   P0
              For example, if the dividend is $10/sher and stock price is $100,
              then market capitalization rate, r, is 0.10.
              Similarly, price equals:

                                     Div1 EPS1 $10. 00
                              P0 =       =    =        = $100
                                      r    r    0.10

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           NOTE: E(r) for growing firms can also equal the earnings-price
           ratio. Key is whether earnings are reinvested to provide a return
           greater or less than the market capitalization rate.
      EXAMPLE (B&M): Effect on stock price of investing an additional $10
      in Year 1 at different rates of return. Project Costs are $10.00 - EPS1.
      NPV = -10 + C/r, where r = 0.10
                                                                Project's Impact
 Project Rate        Incremental           Project NPV          on share Price         Share Price
 of Return           Cash Flow, C          in Year 1            in Year 0              in Year 0, P0        EPS/P0          r
 -----------------   -------------------   ------------------   --------------------   ------------------   -------------   -----
 0.05                $0.50                 -$5.00               -$4.55                 $95.45               .105            .10
 0.10                $1.00                        0                    0               $100.00              .10             .10
 0.15                $1.50                 +$5.00               +$4.55                 $104.55              .096            .10
 0.20                $2.00                 +$10.00              +$9.09                 $109.09              .092            .10
 0.25                $2.50                 +$15.00              +$13.64                $113.64              .088            .10

      NOTE: The earnings-price ratio equals the market capitalization rate (r)
      only when the new project's NPV = 0. Managers frequently make poor
      financial decisions because they confuse earnings-price ratios with the
      market capitalization rate.

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    PVGO - Present Value of Growth Opportunities
    • Think of stock price as the capitalized value of avaerage earnings under

      a no-growth policy, plus PVGO, where
                             P0 = r 1 + PVGO

    Rearranging, we see the earnings-price ratio is equal to:
                                  = r(1− PVGO )
                              P0           P0

      •   Earnings-price ratio will underestimate r if PVGO is positive and
          overestimate it if PVGO is negative.

      Valuation of PVGO
      • We can compute the PVGO using the simplified DCF formulation,

        replacing dividends with forecasted NPV1, where
                             PVGO = r − g1

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           •   For "growth" stocks, PVGO is high and P/E ratio is high (or
               earnings-price ratio is low.
           •   Growth stocks are those which PVGO is large relative to the
               capitalized value of earings per share - not because it's rate of
               growth is high.

           EXAMPLE: P0 = $30, EPS = $1.00, r = 0.10
              1. Capitalized value of earnings equals $10
              2. PVGO = $20.
              3. PVGO is very large relative to the price of the stock.

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    Price-Earnings Ratio (or Multiple)
      •   Market price divided by its current earnings per share.
      •   Measures how much investors are willing to pay for a company with
          that level of earnings.
      •   Importance:
                1. changes in P/E ratio directly impacts stock prices.
                2. current price is just the product of a stock's P/E ratio and EPS
                           Current Price = P/E Ratio x EPS
                3. P/E ratio translates earnings into value; as the P/E ratio
                   expands or contracts, so does price.
                4. P/E ratios tend to accelerate changes (in either direction) as
                   earnings/share changes.

      •   P/E Determinants - in general, P/E's increase when:
                1. Growth rate of earnings are expected to increase.
                2. Dividend payout ratio is expected to increase.
                3. Riskiness of a stock decreases.

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           Three important factors associated with P/E ratios:
               1. Earnings potential
               2. Dividends
               3. Investment quality

           NOTE: Since firms have relatively stable dividend policies, the
            variables most likely to affect P/E are risk and expected

      High P/E Ratio
          • Good growth opportunities

          • Earnings are relatively safe

          • Deserves low capitalization rate (r).

      Low P/E Ratio
         • Low or no growth opportunities

         • Unstable earnings

         P/E Ratios Can Be Misleading

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           •   Firms can have high P/E ratios and almost 0 earnings - generates
               exptremely high P/E ratio.
           •   No reliable association between P/E ratio and the market
               capitalization rate, r; ratio of EPS to P0 measures r only if
               PVGO = 0; EPS must represent earnings with no growth.
           •   Earnings per share means different things to different firms
               because of arbitrary accounting practices, i.e., depreciation
               methods, valuation of inventory, capitalizing vs. expensing, etc.

                 P/E Ratios - Wall Street Journal (10/7/92)

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  Company         P/E Ratio      Company           P/E Ratio
  -------------   ------------   -------------     ------------
  Exxon           17             Coca-Cola         29
  Mobil           23             Hewlett-Packard   14
  Chevron         25             Goodyear          12
  Texaco          17             General Mills     20
  Amoco           25             Phillip-Morris    3
  Shell           16             Paine-Webber      4
  ARCO            32             Shell Oil         16
  Anadarko        97

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Investment Criteria               •   Net Present Value
    •   Discounted/Undiscounted Payback (Payout)
    •   Average Return on Book Value
    •   Internal Rate of Return
    •   Profitability Index

    Net Present Value

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             •    Any investment rule which does not recognize the time
                  value of money is useless.
             •    A dollar today is worth more than a dollar tomorrow.
             •    Depends solely on forecasted future cash flows and
                  opportunity costs of capital
             •    Additivity Rule: Because present values are all measured in
                  units of today's dollars, you can add them up.
                       NPV(A+B) = NPV(A) + NPV(B)

Year        Project 1   Project 2   Project 3    Proj. 1&3    Proj. 1&2   Proj. 2&3
0           -100        -100        -100         -200         -200        -200
1           0           225         450          450          225         675
2           550         0           0            550          550         0
IRR         135%        125%        350%         213%         131%        238%
NPV (10%)   354         105         309          663          459         414

             Payback (Payout)•           Initial investment should be
             recoverable within some specified cutoff period.• Computed by

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              counting the number of years it takes before cumulated cash
              flows equal initial investment.

                                   Cash Flow
 Project   Period 0    Period 1    Period 2      Period 3    Payback        NPV (10%)
 A         -2000       2000        0             0           1              -182
 B         -2000       1000        1000          5000        2              3492

              Payback rule says accept A over B;
              NPV rule says reject A and accept B.
              NOTE: Payback criterion gives equal weight to all cash flows
              before payback date and no weight at all to subsequent cash

                                     Cash Flow

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Project    Period 0    Period 1  Period 2      Period 3     Payback             NPV (10%)
A          -2000       1000      1000          5000         2                   3492
B          -2000       0         2000          5000         2                   3409
C          -2000       1000      1000          100,000      2                   74,867
             Firm accepts too many short-lived projects, possibly with
             negative NPV's.

Choosing a Cutoff Period
             •   Firms typically arbitrarily choose the cutoff period.
             •   If one knows the typical pattern of cash flows, then you can
                 find the optimal cutoff period that maximizes NPV.

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             •   CAUTION: Only works if cash flows are spread relatively
                 evenly over the life of the project.
                   Optimal Cutoff Period = 1 − 1 n
                                           r r (1+ r )
               where n = life of the project
             EXAMPLE: 10% discount rate, 10 year project.
                      So, optimal cutoff is 6 years.

    Discounted Payback
             •   More appropriate because opportunity cost of capital is
                 integrated - but still ignores after payout cash flows.

    Average Return on Book Value
             •   Book rate of return

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             • Divide average forecasted profits (after depreciation and
               taxes) by the average book value of the investment.
             • Ratio is then measured against book rate of return (ROR) of

               the firm as a whole.
             EXAMPLE: Assume $9000 investment, depreciated at $3000
               per year; assume no taxes.
                                            Cash Flow
       Project                  Period 1    Period 2       Period 3     Book ROR
       A         Cash Flow      6000        5000           4000
                 Net Income 3000            2000           1000         44%
       B         Cash Flow      5000        5000           5000
                 Net Income 2000            2000           2000         44%
       C         Cash Flow      4000        5000           6000
                 Net Income 1000            2000           3000         44%
             Average Book Value of Investment:
    Year 0         Year 1         Year 2          Year 3         Average
    $9000          $6000          $3000           $0             $4500

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             NOTE: All projects have 44% book ROR (2000/4500), yet
              Project A clearly has a higher NPV.

             Limitations to Average Return on Book Value:
             • No allowance for immediate cash flows being more valuable

               than distant ones.
             • Not based on cash flows, but accounting income; these are

               often very different due to accounting practices.
             • Yardstick for evaluating average book ROR is arbitrary.

    Internal (or Discounted Cash Flow) Rate of Return
             •   Rule: Accept investment opportunities offering rates of return
                 in excess of the opportunity cost of capital.
             •   Single period model

                        ROR =     Payoff −1

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                       NPV = C0 +         =0
                                     1+ r

             where discount rate that makes NPV = 0 is the internal rate of

                 Year 0               Year 1           Year 2
                 -$4000               $2000            $4000
             so,            NPV = −4000 + 2000 + 4000 2 = 0
                                         (1+ IRR ) (1= IRR )
                IRR = 0.28 (28%)
             NOTE: IRR and the opportunity cost of capital are not the same.

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             •   IRR gives the same answer as NPV rule whenever NPV os a
                 project is a smoothly declining function of the discount rate.

    Pitfalls Associated with the IRR Rule
             1. Lending versus Borrowing Situations

       Project           Period 0          Period 1       IRR         NPV-10%
       A                 -1000             1500           50%         $364
       B                 1000              -1500          50%         -$364
                 •   Lending (A) versusBorrowing (B) -When we lend money, we
                     want a high rate of return; conversely, when we borrow, we want
                     a low rate of return.

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                 2. Multiple Rates of Return
                    • When there is more than one change in the sign of the cash

                      flows, there is generally not a unique internal rate of return.
Project       Period 0       Period 1        Period 2           IRR              NPV-10%
  C            -4000           25,000         -25,000      25% & 400%             -$1,934
                        NPV = −4000 + 25,000 + −25,000 = 0
                                         1.25      (1.25)2
                     NPV = −4000 + 25,000 + −25,000 = 0NOTE: There can
                                     5.0      (5.0)2
                 be as many different IRR's for a project as there are changes in
                 the sign of the cash flows.3. No Internal Rate of Return
    Project       Period 0       Period 1      Period 2       IRR         NPV-10%
     D             1000           -3000          2500          None         $339
                4. Mutually Exclusive Projects
    Project       Period 0        Period 1       IRR            NPV-10%
     E              -10,000        20,000          100%         $8,182
     F              -20,000        35,000          75%          $11,818

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                 NOTE: Maximizing NPV is inconsistent with IRR rule.

    Incremental IRR -
             • Dealing with differences in scale and timing.
             • Evaluate IRR on incremental cash flows from mutually

               exclusive projects.
               • Consider smaller project (E) where IRR is in excess of

                 opportunity cost of capital.
               • Is it worth making an additional investment of $10,000 in


                 Incremental Flows:
   Project           Period 0       Period 1          IRR                NPV-10%
   F-E               -10,000        15,000            50%                $3,636
                 NOTE: Incremental NPV is consistent with additivity rule.

                 1. Differences in Timing of Projects

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     Year              A                     B         B-A
     0                 -23,616               -23,616   0
     1                 10,000                0         -10,000
     2                 10,000                5,000     -5,000
     3                 10,000                10,000    0
     4                 10,000                32,675    22,675
     IRR               25%                   22%       16.6%
     NPV - 10%         $8,083                $10,347   $2,264

                 2. Differences in Scale of Projects
        Year                 X                     Y       Y-X

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        0                  -10,000               -15,000             -5,000
        1                  12,000                17,700              5,700
        IRR                20%                   18%                 14%
        NPV - 10%          $908                   $1089              $181
          • Compare incremental IRR to opportunity cost of capital; if

            greater, take Y instead of X.• Incremental IRR will give same
            investment decisions as NPV.

           3. Other Issues associated with IRR
                • Term structure of interest rates - short-term versus long-

                  term interest rates for short and long-lived projects.
                • Difficult to set benchmark for acceptance of project when

                  opporunity cost of capital changes over time.
                • Project Ranking - should we use IRR? Maximizing NPV

                  subject to capital constraints is better approach.

EB545 – Corporate Finance                                                      94
    Profitability Index
             •    Present value of forecasted future cash flows divided by
                  initial investment.
                                    PI = PV/C0
             •    Rule: Accept all projects with a PI > 1 because this implies
                  they must have NPV >0.
             •    This rule most closely resembles the NPV rule.
             •    Inconsistency in PI Rule

 Project      Period 0           Period 1        PV-10%         PI           NPV-10%
 K            -100               200             182            1.82         $82
 L            -10,000            15,000          13,636         1.36         $3,636
             • Profitability Index suggests that K should be selected over L;

               however, L's NPV is greater than K's.•     Solution: Look at
               P.I. on incremental investment.

Project          Period 0          Period 1        PV-10%          PI            NPV-10%

EB545 – Corporate Finance                                                             95
 L-K          -9,900            14,800         13,454          1.36       $3,554
             • Profitability Index on the additional investment is greater

               than 1, so you know that L is a better project.

EB545 – Corporate Finance                                                          96
Net Present Value Rule and
Investment Decisions
    1. Only Cash flow is Relevant
             •   Dollars received minus dollars paid.
             •   NOT accounting profits.
             •   Estimate cash flows on AFIT basis

    2. Estimate Cash Flows on Incremental Basis
             •   Don't confuse average cash flows with incremental cash
                     EXAMPLE: Decisions about investing in a losing
                     division where "turnaround" opportunities exist should
                     be made based on incremental analysis, not average
             •   Forget sunk costs

EB545 – Corporate Finance                                                     97
                 "spilled milk"
                 past and irreversible outflows
                 cannot be affected by accept/reject decision

             •   Include opportunity costs- cost of a resource may be
                 relevant to the investment decision.
                      EXAMPLE: Decision to construct a new plant on land
                      that could be sold for $1 million (opportunity cost)
                      NOTE: Very difficult to assess when no freely
                      tradeable market exists.

             •   Treat Inflation Consistently
                 Do not confuse nominal and real dollars
                 When using nominal dollars, use nominal discunt rates
                 When using real dollars, utilize real discount rates.

                                        1+ nominal discount rate
                 Real Discount Rate =                            − 1
                                            1+ inf lation rate

    4. Separating Investment and Financing Decisions

EB545 – Corporate Finance                                                    98
             •   Treat all projects as if equity financed.
             •   Treat all cash outflows as coming from stockholders and all
                 cash inflows as going to them.
             •   Caculate NPV, then undertake separate analysis of
                 financing decisions.

    5. Depreciation
             •   Non-cash expense
             •   Important because it reduces taxable income (which are
                 cash flows).
             •   Provides an annual tax shield (product of depreciation and
                 marginal tax rate)
             •   Accelerated depreciation (for IRS) versus straight-line (for
                 stockholders' income statement).

    6. Discounting Inflows and Outflows

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             •   Proper discount rate depends on the risk of the cash flow -
                 not whether it is an inflow or an outflow.
             •   One person's inflow is another person's outflow - in
                 equilibrium, all sellers and buyers must agree on asset
                 values at the margin.
             •   Assumes homogeneous expectations or a well functioning
                 information market.

    7. Choice of Discount Rate
             •   We are only concerned with rates that are available in the
                 capital market today.
             •   What is the value of $1 discounted at today's 1-year
                 interest rate (same for n years).

EB545 – Corporate Finance                                                      100
Other Capital Budgeting Issues
    Optimal Timing of Investment

             •   The fact that a project has a positive NPV does not mean
                 that it is best undertaken now.
             •   Negative NPV projects today might become a valuable
                 opportunity at some later date.
             •   Examine alternative dates and find the one which adds the
                 most to the firm's current value.

                            Net future value as of date t
                                       (1+ r)t

             EXAMPLE: Timber cutting (B&M)

EB545 – Corporate Finance                                                    101
Year of Harvest         0        1           2          3          4           5
Net Future Value        50    64.4        77.5        89.4      100          109.4
                  NPV if harvested in Year 1 = $64.4/1.10 = $58.5
Year of Harvest         0        1           2          3         4             5
Net Present Value     50       58.5       64.0       67.2       68.3          67.9
    Optimal Point of Investment is Year 4 - where NPV is maximized.

    Equipment Utilization and Replacement
              Weigh increased costs of utilization against cost of replacement.

EB545 – Corporate Finance                                                            102
             Equivalent annual cost: represents an annuity with the same PV
             and life as the equipment or machine.
                  PV of Annuity = PV of Cash outflows for Machine
                                  = annuity payment x t -year annuity factor

        EXAMPLE: The president's executive jet is not fully utilized. You
        judge that its use by other officers would increase direct operating
        costs by only $20,000 per year and would save $100,000 a year in
        airline bills.A new jet costs $1.1 million and (at its current low rate
        of use) has a life of 6 years. Assume the opportunity cost of capital is
        12% and the company pays no taxes. Should you try to persuade the
        president to allow other officers to use the plane.

             With a 6 year life the equivalent annual cost (at 12%) of a new
             jet is $1,100,000/4.111 = $267,575. If the jet is replaced at the
             end of year 3 rather than 4, the company will incur an

EB545 – Corporate Finance                                                          103
              incremental cost of $267,575 in Year 4. The present value of
              this cost is:
                            $267,575/1.124 = $170,049

                     The present value of the savings is:
                                   ∑ 112t = $192,147
                                  t =1
                     So, the president should allow wider use of the present jet
                     because the PV of the savings is greater than the PV of the

        NOTE: As an aside, although a decision on usage of the new jet can be postponed, it
        appears that other officers should continue to use the plane. If the increased usage causes
        the plane life to be reduce 25% (i.e. 6 to 4.5 years), the increase in the equivalent annual
        cost is $330,420-267,548 = $62,872. This is less than the annual saving of $80,000.

    Equipment Replacement
              1.     Present Value of Incremental Cash Benefits

EB545 – Corporate Finance                                                                              104
             Consider the following:
                 CF = Cash flow
                 ∆ = New minus Old
                 S = Sales
                 C = Operating Costs
                 D = Depreciation
                 t = tax rate

                            ∆CF = (∆S - ∆C - ∆D)(1-t) + ∆D

             2.   Salvage Value
                  •   New machine
                  •   Old machine

             3.   Initial Capital Investment
                  •     Proceeds from sale of old machine

EB545 – Corporate Finance                                    105
                 •    Tax consequences of sale or disposal of old machine
                 •    Cost of new machine

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Capital Rationing
    •   Basic assumption is that capital is limited.
    •   Inconsistent with notion of perfect capital
    •   "Soft" rationing vs. "hard" rationing
        •    Soft Rationing
             •   Provisional and arbitrary limits set by management of the
             •   Not indicative of any real imperfections in capital markets.
             •   Means of controlling investment among managers.
             •   Means of dealing with biased cash flow forecasts.
             •   For projects with significant NPV's, firm is capable of
                 raising more money and loosening the artificial constraint.

EB545 – Corporate Finance                                                       107
        •    Hard Rationing
             •   Wealth of firm's shareholders is maximized when the firm
                 accepts every project that has a positive net present value.
             •   Refleccts a barrier between the firm and capital markets -
                 market imperfections
             •   Shareholders lack free access to well-functioning capital
             •   Projects with positive NPV's are passed up
             •   Often there are other issues at work, i.e. control of the firm.
             •   Must not ignore risk of the project - opportunity cost of
                 capital takes this into account.

    Capital Rationing Models
             •   Assume there are capital budget limitations.
             •   Must incorporate a method of selecting among the
                 available opportunities that maximizes NPV of the firm.

EB545 – Corporate Finance                                                          108

         EXAMPLE (B&M): Suppose there is a 10% opportunity cost of
            capital, company has total resources of $10 million and has the
            following opportunities:

                              Cash Flow
                              Millions $                  NPV - 10%
      Project    Period 0     Period 1      Period 2      $ Millions      P.I.
      A          -10          +30           +5            21              3.1
      B          -5           +5            +20           16              4.2
      C          -5           +5            +15           12              3.4
         •    Firm has sufficient resources to invest in either project A or in
              projects B&C.
         •    Individually, B&C have lower NPV's - but together they have a
              greater NPV than A.
         •    Choosing among projects based solely on individual NPV's may
              lead to sub-optimal decision.

EB545 – Corporate Finance                                                         109
        •    Alternative: Select projects that offer the highest ratio of PV to
             initial outlay (PI).

                  Profitability Index = Present Value

        Limitations to PI:
            1. May only be applicable to a single period model.
            2. When there are other constraints on the choice of projects
            3. Unable to utilize when there are project dependencies.

        EXAMPLE (B&M):
                               Cash Flow

EB545 – Corporate Finance                                                         110
                               Millions $                 NPV - 10%
        Project   Period 0    Period 1      Period 2       $ Millions      P.I.
        A          -10         +30           +5            21              3.1
        B          -5          +5            +20           16              4.2
        C          -5          +5            +15           12              3.4
        D          0           -40           60            13              1.4

        •    If we accept B&C, we cannot accept D which is more than
             budget limit in Period 1.
        •    However, we can select A in Period 0 and have sufficient cash
             to take D in Period 1.
        •    NOTE: A&D have lower PI's than B&C -but have a higher
        •    Because constraint applies to two periods, PI index fails us as an
             appropriate capital rationing criteria.

        2.   Linear Programming Models
             •   General optimization procedure.

EB545 – Corporate Finance                                                         111
              •       Mathematical model with the folowing properties:
                      1.   A linear objective function which is to be maximized or minimized.
                      2.   A set of linear constraints.
                      3.   Variables which are all restricted to nonnegative values.

              The Ice-Cold Refrigerator Company can invest capital funds in
              a variety of company projects which have varying capital
              requirements over the next 4 years. Faced with limited capital
              resources, the company must select the most profitable projects
              and budgets for the capital expenditures. The estimated present
              values of the projects, the capital requirements and the available
              capital projections are shown below:

                               Present                    Capital       Require.
    Project                    Value         Year 1       Year 2        Year 3         Year4
    Plant Expansion            90,000        15,000       20,000        20,000         15,000

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    Warehouse Expansion     40,000     10,000    15,000     20,000     5,000
    New Machinery           10,000     10,000    0          0          4000
    New Product Research    37,000     15,000    10,000     40,000     35,000

    Available Capital                  40,000    50,000     40,000     35,000
    The following definitions are chosen for decision variables:
    •    x1 = 1 if the plant expansion project is accepted; 0 if rejected.
    •    x2 = 1 if the warehouse expansion project is accepted; 0 if rejected.
    •    x3 = 1 if the new machinery project is accepted; 0 if rejected.
    •    x4 = 1 if the new product research project is accepted; 0 if rejected.

    Linear Programming formulation has:
    1. Separate constraint for each year's available funds;
    2. Separate constraint requiring each variable is less than or equal to 1.

EB545 – Corporate Finance                                                         113
    3.   Integer Programming Models
         •    Applicable to real-world problems where accept/reject decisions
              must be made.
         •    Method avoids fractional values in solution set.
         •    Involve 0-1 or binary integer variables.
         •    Widely use method in capital budgeting problems.

    4.   Multi-Objective Decision Models
         •   Utility theory based model - applicable in a world of certainty
             or uncertainty.
         •   Means of incorporating multiple investment criteria
         •   Profitability index, NPV, IRR, Payout, etc.

EB545 – Corporate Finance                                                       114
        •    Evaluate firm's relative strength of desirability on each
        •    Evaluate relative importance of each criteria (set of weights).
        •    General mathematical form:

EB545 – Corporate Finance                                                      115
Risk and Return
    •   "Opportunity cost of capital depends on the risk
        of the project"
    •   Major topics of interest
             Security Risk
             Portfolio Risk

    •   Risk-return relationships in capital market over
        the last 60 years
             1.   Common stocks - (S&P composite index)
             2.   Government bonds
             3.   Corporate Bonds
             4.   Treasury Bills

EB545 – Corporate Finance                                  116
                   Average Annual Rate of Return
                         Nominal     Real            (versus T-Bills)
                         ROR         ROR             Risk Premium
    Treasury Bills       3.6%        0.5%            0
    Government Bonds 4.7%            1.7%            1.1%
    Corporate Bonds      5.3%        2.4%            1.7%
    Common Stocks        12.1%       8.8%            8.4%

                        (Note: Average inflation - 3.1%)
    1. Treasury Bills: short maturity and no risk of default (inflation risk).
    2. Government Bonds: longer term asset, sensitive to changes in interest rates
    (bond prices fall when interest rates rise and vice versa).
    3. Corporate Bonds: similar to government bonds, except risk of default.
    4. Common Stocks: direct share in the risk of the enterprise.

    •   Evaluating and investment opportunity with
        same risk as S&P 500 - or market portfolio.

EB545 – Corporate Finance                                                            117
             Hypothesis: Opportunity cost or discount rate should be E(r) on
             market portfolio (rm).
             Assumption: Future is like the past and we use a 12.1% ROR.
             Methodological Shortcoming: Market returns are not stable
             over time.
             Alternative: Look at the individual components of rm
                 a. risk-free rate (rf)
                 b. risk premium
                 rm (1990) = rf (1990) + normal risk premium
                             = 0.08 + 0.084
                             = 0.164 (16.4%)

             (NOTE: This assumes a normal, stable risk premium)

    •   Evaluating and investment opportunity with risk
        characteristics different from S&P 500.

EB545 – Corporate Finance                                                      118
    •   Measures of Risk and Return
             1.   Expected Return: a measure of central tendency.
                       E (r ) = ∑ hri
                               i =1
             where h = probability of occurrence of return i

             2.   Variance: An average deviation meaure of dispersion,
                  obtained by averaging the squares of deviations of
                  individual observations from the mean.
                       σ2 (r ) = ∑ hi[ri − E (r )]2
                                i =1

             3.   Standard Deviation: an average deviation measure of
                  dispersion equal to the positive square root of the variance.
                                   σ2 = σ

             4.   Covariance: an absolute measure of the extent to which
                  two sets of variables move together over time.

EB545 – Corporate Finance                                                         119
                                ∑[(rat − ra )(rb − rb )]
                       Covr r = t =1            t           (sample covariance)
                           a b          n −1

                   1        2           3       4          5      Mean
       Stock A     0.04     -0.02       0.08    -0.04      0.04   0.02
       Stock B     0.02     0.03        0.06    -0.04      0.08   0.03

              (0.04-0.02)(0.02-0.03)        = -0.0002
              (-0.02-0.02)((0.03-0.03)      = 0.0000
             (0.08-0.02)(0.06-0.03)         = 0.0018
             (-0.04-0.02)(-0.04-0.03)       = 0.0042
             (0.04-0.02)(0.08-0.03)         = 0.0010

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                                   Total       0.0068

                      Covr r = 0.0068 = 0.0017
                            a b (5−1)
             •   Positive Covariance: when one stock produces a return
                 above its mean, the other tends to do as well.
             •   Note: Covariance number tells us little about the
                 relationship between returns on the two stocks.

        •    Population Covariance
                      Covr r = ∑ hi[ra − E (ra )][rb − E (rb )]
                            a b
                                           i        i
                                  i =1
        5.   Correlation Coefficient: normalized measurement of joint
             movement between two variables.
             •   Covariance number is unbounded (- to + infinity)
             •   Put bounds on covariance by dividing by the σ of the two
             •   Correlation coefficient, ρ, ranges from +1 to -1.

EB545 – Corporate Finance                                                   121
                            ρa , b = σ σ ra b

                                      r r
                                      a    b

EB545 – Corporate Finance                       122
                                Correlation Coefficient
               r                                                  r
                   b                                                  b

                            r                                             r
                              a                  Perfect negative             a
                       Perfect Positive           Correlation
                        Correlation                  (-1)


                                Uncorrelated                  a

EB545 – Corporate Finance                                                         123
        •    Note that in the case of perfect positive and negative
             correlation, if we know the returns on stock b, then we could
             predict a.
        •    Magnitude of the slope is irrelevant.
        •    Coefficient may range from -1 to +1.

        Also, it follows that:
                            Covr r = ρa , bσr σr
                                 a b        a   b

        6.   Characteristic Line'
             •  Extension of covariance to relationship between individual
                stock and market portfolio.

             Month 1        Month2          Month 3    Month 4        Month 5
Stock J      0.02           0.03            0.06       -0.04          0.08
Market Port. 0.04           -0.02           0.08       -0.04          0.04

EB545 – Corporate Finance                                                       124
                                          CHARACTERISTIC LINE



                                              Slope of Line = Bj
     j's Return


                  -0.04   -0.02           0            0.02        0.04   0.06   0.08



                                                 Market Return

EB545 – Corporate Finance                                                               125
        •    Characteristic Line

                 1.   Describes the relationship between returns on the
                      stock and the market portfolio.
                 2.   Shows the return that you expect the stock to produce,
                      given a particular market ROR.
                 3.   The slope of the characteristic line is the stock's beta
                      factor or βj (refer to the intercept of the characteristic
                      line as a.)

        7.   Beta                               Intercept

EB545 – Corporate Finance                                                          126
                    Cov(rj ,rm )
             Bj =                                 a j = r j − β jrm
                      σ2r  m

                                                                (rm, t − rm )2
                    Variance of market returns:         σ =∑
                                                           t =1    n −1
             From preceding Table, market σ2 = 0.0024;
                 βj = 0.0017/0.0024 = 0.708
                 aj = 0.03-0.708(0.02) = 0.0158

        •    Beta is an indicator or the degree to which a stock responds
             to changes in the return produced by the market.
        •    If we knew the market was going to be higher by 1% next
             month, we would increase our expectation for stock j's
             return by 0.708%

    Reducing Risk Through Diversification

EB545 – Corporate Finance                                                        127
             •   Most stocks are substantially more variable than the market
                 portfolio - only a few have lower standard deviation of
             •   Why doesn't the market's variability reflect the average
                 variability of its components?
             •   Diversification works to reduce risk because prices of
                 different stocks do not move exactly together.
             •   Even in a 2 stock portfolio, a decline in the value of one stock
                 in the portfolio can be offset by a rise in the price of the other
             •   Result: Variability of portfolio is less than the average
                 variability of the two stocks in the portfolio.

EB545 – Corporate Finance                                                             128

                             1        5          10          15          Number of

    Systematic versus Unsystematic (unique) Risk
             1.       Unsystematic Risk (unique)
                  •   Risk that can be potentially eliminated by diversification.
                  •   Firm or industry specific.
                  •   Management Risk
                  •   Financial or operating leverage
                  •   Able to diversify away.

EB545 – Corporate Finance                                                              129
             2.       Systematic Risk
                  •   Stems from economy wide perils which affect all
                  •   Interest rate risk
                  •   Purchasing power risk (inflation)
                  •   Stock market risk
                  •   Unable to completely diversify away.
                  •   For a reasonably diverisfied portfolio, this is the only risk
                      that matters.

EB545 – Corporate Finance                                                             130
    Portfolio Analysis
               •   Expected Return: weighted average of the expected returns
                   on the individual stocks contained in the portfolio.
                                E (rp ) = ∑ x j E (r j )

                                           j =1

                   where, xj is the percent of investment in stock j

           •   Variance: function of the variance of each stock included in the
               portfolio as well as the covariance between stocks' returns.
                          σ2 = xi2σi2 + x2σ2j + 2 xi x jρi , jσi σ j
                           p             j

                        xi = % of portfolio invested in asset i
                       σi2 = variance of returns on asset i
                       ρij = correlation coefficient between assets i & j
                       where -1 ≤ ρij ≤ 1

EB545 – Corporate Finance                                                         131
      Example of Portfolio Effect:
                Return & Probabilities for investments 1 and 2
   State of Economy     Probability      1's Return     2's Return
   Recession            0.20             5%             20%
   Slow Growth          0.30             7%             8%
   Moderate Growth      0.30             13%            8%
   Strong Economy       0.20             15%            6%
              Expected Value = E(r1) = ∑ r1 p1 = 10.0%

              Expected Value = E(r2) = ∑ r2 p2 = 10.0%

          Standard Deviation = σ1 = ∑ (r1 − r1)2 p1 = 15.4% = 3.9%
          Standard Deviation = σ2 = ∑ (r2 − r2 )2 p2 = 25.6% = 51%
                                                                .    PORTFOLIO
                       E (rp ) = ∑ x j E (r j )

                                  j =1

                             E(rp) = 10%

EB545 – Corporate Finance                                                   132
                 Possible Outcomes




                        Recession      Slow          Semi-    Strong
                                       Growth        Strong   Growth
                 σ2 = xi2σi2 + x2σ2j + 2 xi x jρi , jσi σ j
                  p             j

        where,   x1 = 0.50, σ1 = 3.9
                 x2 = 0.50, σ2 = 5.1
                 ρ1,2 = -0.70
                 σp = 1.8%

EB545 – Corporate Finance                                              133
      •   Portfolio standard deviation is less than the standard deviation of
          either investment (3.9% or 5.1%).
      •   Any time two investments have a correlation coefficient less than
          +1, some risk reduction will be possible by combining the assets
          in a portfolio.
      •   The extent that we can still get risk reduction from positively
          correlated items gives extra meaning to portfolio management.

                 Impact of various assumed correlation coefficients for
                   Investments 1 and 2:

                    Correlation Coefficient Portfolio Stand. Dev.
                        +1.0                     4.5%
                        +0.5                     3.9%
                         0.0                     3.2%
                        -0.5                     2.3%
                        -0.7                     1.8%
                        -1.0                     0.0%

EB545 – Corporate Finance                                                       134
            State Probability RA   RB
            1     0.25        0.06 0.12
            2     0.25        0.08 0.10
            3     0.25        0.10 0.08
            4     0.25        0.12 0.06

      E(RA) = 0.09                          Var(RA) = 0.0005
      E(RB) = 0.09                          Var(RB) = 0.0005
      Cov(RA,RB) = ∑ Pi[ RA − E ( RA )][ RB − E ( RB )] = −0.0005

                        i =1

      Correlation Coefficient =
        ρ A, B =
                   COV ( RA RB )
                                 =       −0.0005      = −1
                     σ Aσ B        ( 0.0005)( 0.0005)
      Portfolio with 50% in A and 50% in B:
                  E(Rp) = 0.09         Var(Rp) = 0

EB545 – Corporate Finance                                           135
      •   In this portfolio the expected return is a certain return. No
          matter which state occurs, the return is 0.09.
      •   If there is a perfect negative correlation between two assets, then
          the Var(Rp) equals 0. This means it is a risk-free portfolio.

          Alternatively, if perfect positive correlation exists between two
          assets, the portfolio standard deviation is a weighted average of
          the assets' standard deviation.
          Asset A: E(R) = 0.10          σ = 0.05       xA = 0.50
          Asset B: E(R) = 0.12          σ = 0.08       xA = 0.50

                   Assuming ρA,B = 1.0, then σp = 0.065


EB545 – Corporate Finance                                                       136
          1. Risk of a well-diversified portfolio depends on the market risk of
             the securities included in the portfolio.
          2. In other words, how sensitive are each of the stocks in the
             portfolio to market movements, both individually and
          3. Stocks with β greater than 1.0 tend to amplify overall movements
             in the market - both upward and downward.


EB545 – Corporate Finance                                                         137
           1. The variabilityof a portfolio reflects mainly the covariance.
           2. When there are just two securities in a portfolio, there are an
              equal number of variance and covariance boxes.
           3. As the number of stocks, n, increases in the portfolio, you
              steadily approach the average covariance.
           4. If average covariance were zero, it would be possible to
              eliminate all risk; however, covariances are almost always
           5. No matter how many stocks you include, there are limits to the
              benefits of diversification.

EB545 – Corporate Finance                                                       138
Maximizing Return Given Risk
    The Minimum Variance and Efficient Sets
       (Markowitz Model)
             1. Given the level of risk, or standard deviation, we prefer
                portfolio positions with higher expected rates of return.
             2. Given the level of expected return, we prefer portfolio
                positions with lower risk.
             3. Given the characteristics of the available population of stocks,
                the investment opportunity set has a perimeter that is
                represented by a bullet-shaped curve.
             4. This perimeter is referred to as the minimum variance set.

EB545 – Corporate Finance                                                          139
                 Return (%)





                              5   10   15   Standard
                                            Deviation (%)

EB545 – Corporate Finance                                   140
         • Each point on the minimum variance set represents a portfolio,

           with portfolio weights allocated to each of the stocks in the
         • Given a particular expected rate of return, the portfolio on the

           MV set has the lowest possible standard deviation achievable
           with the available population of stocks.
         • The terms minimum variance set and minimum standard

           deviation set can be used interchangeabley.
         • The minimum variance set can be divided into two halves, a top

           and bottom - separated at the minimum variance point (MVP).
         The MVP represents the single portfolio with the lowest possible
           standard deviation, the global minimum variance portfolio.
         • The most desirable portfolios for us to hold are those in the top

           half of the bullet.
         • Top half of the bullet is call the efficient set - Given a

           particular level of standard deviation, the portfolios in the
           efficient set have the highest attainable expected rate of return.

EB545 – Corporate Finance                                                       141
    Finding the Minimum Variance Portfolio
              Consider an example where we build portfolios from three
              available stocks with the following expected returns:
                        A, Acme Steel:                5%
                        B, Brown Drug:                10%
                        C, Consolidated Electric:     15%
              The covariance matrix for the stocks is given by:
                                A        B        C
                      A         0.25 0.15         0.17
                      B         0.15 0.21         0.09
                      C         0.17 0.09         0.28
          By taking the square root of the variances going down the diagonal
  of the matrix, we can compute the standard deviations of the stocks as
                        A, Acme Steel σ(rA):               0.50
                        B, Brown Drug σ(rB):               0.46
                        C, Consolidated Electric σ(rC):    0.53

EB545 – Corporate Finance                                                      142
                Return (%)

                             MINIMUM VARIANCE SET FOR
                             CONSOLIDATED, BROWN & ACME

                                 MVP         Consolidated


                            10 20 30 40 50 60    Standard
                                                Deviation (%)

EB545 – Corporate Finance                                       143
                   Portfolio Weight
                    in Stock B xB




                              S                  T
                -1.0          0
                                           1.0       Portfolio Weight
                                                      in Stock A


EB545 – Corporate Finance                                          144
           •   Acme Steel (A) and Brown Drug (B) portfolio weights are
           •   Consolidated Electric's portfolio weight is implied by the value
               for xA and xB
           •   For example, if we are at Point U, portfolio weight in Brown is
               equal to 0.30 and portfolio weight in Acme is 0.0. Since there
               are only three stocks, portfolio weight in Consolidated is equal
               to 1- xA - xB = 0.70.
           •   Each point represents a portfolio with particular weights
               assigned to each stock.

           DEF. "Selling Short": When you sell short, you borrow shares of
            stock from someone (usually through your broker). You are
            obligated to return to this person, after a certain period of time,
            the same number of shares you borrowed. When short-selling a
            stock as part of your portfolio, the weight assigned to the stock
            is negative.

EB545 – Corporate Finance                                                         145
      Now consider the triangle area bounded by points R,S & T.:
         1. All points in the triangle represent portfolios where we have
            invested positive amounts in each of the three stocks.
         2. Consider Point L, where we invest 50% in Acme, 45% in Brown
            and 5% in Consolidated.
         3. For portfolios on the perimeter, we invest a combined total of
            100% of our money in two stocks and 0 position in the third.
         4. At point Q, we invest 60% in Brown, 40% in Conslolidated and
            nothing in Acme. Conversely, on perimeter RT we are taking no
            position in Consolidated.
         5. If we are outside the triangle, at any point to the northeast of the
            line Y'X, we are selling Consolidated short.
         6. If we are positioned to the west of the figure's vertical axis, we
            are selling Acme short.
         7. If we are anywhere to the south of the horizontal axis, we are
            selling Brown short.
         8. At point Y', we are selling Acme short and adding the proceeds
            to our equity to invest in Brown.

EB545 – Corporate Finance                                                          146
           Iso-expected Return Lines: Line, drawn on a mapping of
             portfolio weights, which shows the combinations of weihts all of
             which provide for a particular portfolio rate of return.
                  •   In general, the slope and relative position of the iso-expected return lines are
                      dependent on the relative expected returns of the stocks considered.
                  •   If we want to see combinations of weights that are consistent with a different value
                      for portfolio return, then we must look at different iso-expected return lines.
           Iso-variance ellipse: Ellipse, drawn on a mapping of portfolio
             weights, which shows the combinations of weights all of which
             provide for a particular portfolio variance.
                  •   Ellipses are all concentric about the point MVP.
                  •   Similar to lines denoting points of equal altitude on a topographic map.
                  •   Iso-variance ellipses represent points of constant variance.
           Critical Line: A line tracing out, in portfolio weight space, the
             portfolio weights in the minimum variance set.
                  •   Critical line can be found by tracing out points of tangency between iso-expected
                      return lines and iso-variance ellipses.
                  •   Finding the minimum variance set is tantamount to finding the location of the critical

EB545 – Corporate Finance                                                                                 147
                    Portfolio Weight
                     in Stock B     x



             -1.0                 0                               1.0
                                                                        Portfolio Weight
                                                                          in Stock A

                                                      12%   10%
                                -1.0            16%

EB545 – Corporate Finance                                                                  148
    Portfolio Theory with Borrowing and Lending
             •   Suppose that you can lend and borrow money at some risk-
                 free interest rate, rf.
             •   For example: Invest some of your money in T-Billls and
                 some in common stock portfolio M.





EB545 – Corporate Finance                                                   149
             Since borrowing is negative lending, you can borrow at rf
             and invest more in stock portfolio M.

             NOTE: What we see is that regardless of the level of risk
             preferred, you get the highest expected rate of return by a
             mixture of portfolio M and borrowing or lending.
             1. The best portfolio of stocks must be selected and that is
                the portfolio where the capital market line is tangent to
                the efficient frontier.
             2. Blend this portfolio with borrowing and lending to match
                individual risk preferences.

EB545 – Corporate Finance                                                   150
    Capital Asset Pricing Model
             1. All investors seek to invest in Markowitz efficient portfolios.
             2. Unlimited borrowing and lending at the risk-free rate.
             3. All investors have homogeneous expectations.
             4. No taxes and no transaction costs.
             5. Capital markets are in equilibrium.

               Central Message: In a competitive market, the expected
               risk premium varies in direct proportion to beta.

EB545 – Corporate Finance                                                         151
                                            SECURITY MARKET LINE


                              A                         Market Portfolio

                               0.5        1.0              2.0Beta
Expected Risk Premium on investment = (Beta) x (Expected Risk Premium on Market)
                              ri - rf = βi(rm-rf)
So, the expected risk premium on portfolio A is 1/2 the risk premium on the market, and
portfolio B has twice the expected risk premium.

EB545 – Corporate Finance                                                           152
             In terms of Beta:
                   E (ri ) = rf + βi[ E (rm − r f )]
                               Cov(ri ,rm )
                        βi =
                                σ2 (rm )

             So, for capital budgeting purposes, where the risk
             characteristics of the project are consistent with the risk of
             firm i, the the opportunity cost of capital for further
             investment is E(ri).

EB545 – Corporate Finance                                                     153
    Four Basic Principles of Portfolio Selection:
             1. Investors like high expected returns and low standard
                deviation; stock portfolios that offer highest E(r) for a given σ
                are efficient portfolios.
             2. To know the marginal impact of a stock on a portfolio, you
                must look at its contribution to portfolio risk; contribution
                depends on the stock's sensitivity to changes in the value of
                the portfolio.
             3. Stock's sensitivity to changes in the value of market portfolio
                is Beta (β) - measures marginal contribution of a stock to risk
                of the market portfolio.
             4. If investors can borrow and lend at the risk-free rate of
                interest, then they should always hold a mixture of the rf
                investment and one particular common stock portfolio.

EB545 – Corporate Finance                                                           154
                     LIE ON THE SECURITY MARKET LINE?

                                   M           B
                                               Market Portfolio

              rf         A

                            0.5    1.0             2.0

EB545 – Corporate Finance                                         155
        LINE (SML)?
             1. Would you buy stock A? If you wanted an investment with a
                Beta of 0.50, NO! If every rational investor shares your view,
                then the price of A drops and E(r) goes up.
             2. What about stock B? Again, one can achieve higher return
                (with equivalent risk) by investing in a stock or portfolio on
                the SML. Again, price of B must fall until E(r) is equivalent.

             1. Assume initially, that stock A's expected dividend and
                expected market price are $5.00 and $100 respectively. The
                beginning market price is $100 and consequently, the
                expected rate of return is equal to:

             E (rA ) = E (dividend + ending market price) −1
                             beginning market price

EB545 – Corporate Finance                                                        156
                   E (rA ) = E ($5.00 + $100) −1 = 0.05 (or 5%)

      2. Now suppose an item of good news is received by the market place
         and the information does nothing to change the market's assessment
         of the risk characteristics of the stock; however, it does change its
         assessment of the dividend and ending price.

      3. The new expected dividend is $7.00 and the new expected ending
         price is $105.00. If there is no change in the beginning price, the new
         expected rate of return is:
                   E (rA ) = E ($7.00 + $105) −1 = 0.12

      4. It is now in everyone's interest to increase their investment in stock
         A. The buying pressure on the stock will force its price up and its
         expected rate of return down, until it resumes its equilibrium
         position. The market price at that point will be:

EB545 – Corporate Finance                                                          157
                        0.05 = E ($7.00 + $105) −1
      •   NOTE: This same kind of market pressure holds each stock in its
          equilibrium position. Should anything disturb the stock from its
          position, opportunities will arise to earn superior returns by
          either buying or selling it short. The buying or selling pressure
          immediately forces the stock back to its equilibrium price.

         1. In equilibrium, all stocks must lie on the security market
         2. An investor can always obtain an expected risk premium of
            β(rm-rf) by holding a mixture of the market portfolio and the
            risk-free investment.
         3. Stocks on average lie on the security market line; so, if none
            lie on the line then none can lie above the line.
    Validity and Role of CAPM

EB545 – Corporate Finance                                                     158
  •   While the capital asset pricing model is not perfect, it does seem to
      capture important elements of risk.
  •   There is an important role in this theory of the market portfolio. For any
      efficient portfolio, the expected risk premium on each stock (ri-rf) must be
      proportional to its beta measured relative to that efficient portfolio.
  •   If the market portfolio is efficient, then (ri-rf) will be proportional to beta
      measured relative to the market portfolio.
  •   The market portfolio should in principle include all assets; the available
      market indexes clearly fall short of that ideal.
  •   If a particular index is, ex post, part of the efficient set, then returns and
      beta will plot along a straight line; the problem is how close is close
      enough in terms of fitting a line.
  •   Beta is, however, a valuable measure of risk: simple, objective, plausible
      and consistent with theory - it correctly excludes diversifiable risk.
  •   Regard CAPM as a useful rule of thumb - not as the final word on
      how the world works.

      Arbitrage Pricing Theory (APT)

EB545 – Corporate Finance                                                               159
             • Fundamental assumption is that security returns are generated
               by a process identical to a single index (CAPM) or multi-
               index (Markowitz) model.
             • APT suggests that the covariances that exist between security

               returns can be attibuted to the fact that securities respond, to
               one degree or another, to the pull of one or more factors.
             • Basic assumption is that each stock's return depends partly on

               macroeconomic influences and partly on firm-specific factors.
             • Factors may include oil prices, interest rates, level of

               industrial activity, inflation rate, etc.
             • Assume that the relationship between security returns and

               factors is linear.
                       rj = aj + β1,jI1,j + β2,jI2,j +. . .+ βn,jIn,j
             where,     I = value of any one of the indices which affects the
                        rate of return of the security.
                       β = individual beta for each index (may be positive or

EB545 – Corporate Finance                                                         160
             •   The portfolio's Beta with respect to any one of the factors is a
                 simple weighted average of the betas of the securities in the
                                   β1, P = ∑ x jβ1, j

                                           j =1

             •   As in the CAPM, diversification does eliminate unique or
                 business risk and diversified investors can ignore it when
                 deciding to buy or sell a stock.
             •   APT states that the expected risk premium on a stock should
                 depend on the expected risk premium associated with each
                 factor and the stock's sensitivity to each factor.

EB545 – Corporate Finance                                                           161
             •  If we assume that Beta for each index is 0, then the expected
                risk premium is equal to 0; the portfolio is essentially risk-
                free and should offer a return equal to the risk-free rate.
             • If the portfolio offered a higher return, investors could make a

                risk-free (or "arbitrage") profit by borrowing to buy the
             • If the portfolio offered a lower return, investors could sell the

                portfolio and invest the funds in U.S. Treasury Bills.
             If the arbitrage pricing relationship holds for diversified
                portfolios, it must generally hold for the individual stocks.

EB545 – Corporate Finance                                                          162
    APT vs. CAPM
             •   Similar in that both theories stress that E(r) depends on risk
                 stemming from economy-wide influences and is not affected
                 by unique risk.
             •   If each of APT's factors is proportional and cumulative to
                 CAPM's market portfolio beta, then result is equivalent.
             •   Identifying the factors of importance in APT is difficult.
             •   APT must have (1) reasonably short list of macroeconomic
                 factors; (2) risk premium measures associated with each of
                 these factors; and (3) measurement of stock sensitivity to each

EB545 – Corporate Finance                                                          163
Capital Budgeting Under Uncertainty
    Company Cost of Capital
             •   Rate of return required by investors;
             •   Opportunity cost of capital for firm's existing assets.

    Project Beta
             •   Measure of risk for a specific project or investment
             •   Relate risk of project to market risk.
             •   Difficult to assess.

EB545 – Corporate Finance                                                  164

                                            Cost of Capital

         r f (7%)

                             1.0 1.30
                                                  Project Beta

            Average Beta of Firm's Assets = 1.30
             Note: True cost of capital depends on the use to which the
             capital is put.

EB545 – Corporate Finance                                                 165
               • Accept any project that more than compensates for
                 project beta.
               • Emphasis is on project beta - not firm beta!

               • Note Difference between company cost of capital rule and

                 project beta rule:
                 1. Company cost of capital rule say accept any project with
                    E(r) greater than 15% (above horizontal line).
                 2. Project Beta rule would exclude certain projects above the
                    company cost of capital line.
           •   Project Beta:
                    E(project return) = rf + (project Beta)(rm - rf)
                      State     Prob. rm           r1       r2
                      1         0.10      -0.15 -0.30       -0.09
                      2         0.30      0.05     0.10     0.01
                      3         0.40      0.15     0.30     0.05
                      4         0.20      0.20     0.40     0.08


EB545 – Corporate Finance                                                        166
             Risk-free rate = 0.05              E(r1) = 0.20      E(r2) = 0.03
             E(rmarket) = 0.10                  σmarket = 0.10

    Capital Structure and Company Cost of Capital
             •   Cost of capital is a hurdle rate that depends on business risk
                 of firm's investment opportunity.
             •   But, shareholders also bear financial risk to the extent that the
                 firm issues debt.
             •   The more a firm relies on debt financing, the riskier its
                 common stock.
             •   Borrowing creates financial leverage.
                   Pushes up risk of common stock.
                   Leads to higher E(r)

EB545 – Corporate Finance                                                            167
    Weighted Average Cost of Capital (WACC)
             Definition: Expected return on a portfolio of all the firm's
             securities. Used as a hurdle rate for capital investment.
             WACC = (Aftertax cost of debt x % of debt in capital structure)
             + (Cost of Preferred x % of Preferred in Capital Structure) +
             (Cost of common x % of common in capital structure)

             EXAMPLE:Consider you own 100% of firm A's debt and
             equity. What is your E(r) on this portfolio?
             Company Cost of Capital = rassets = rportfolio
                                =       debt      rdebt + equity requity
                                   debt + equity         debt + equity

                 Asset Value = $100       Debt Value = $40
                                          Equity Value = $60
                 ---------------------    --------------------
                 Asset Value = $100 Firm Value = $100
                 rassets = debt rdebt + equity requity
                            value       value

EB545 – Corporate Finance                                                      168
             If you demand 8% return on debt and 15% return on equity,
                       ACC = 40 (0.08) + 60 (0.15) = 0.122 or 12.2%
                             100         100
    Effect of Changes in Capital Structure
             Assume the firm issues an additional $10 of equity and uses it to
             payoff debt.
             Balance sheet changes to the following:
                   Asset Value = $100 Debt Value = $30
                                         Equity Value = $70
                   --------------------- --------------------
                   Asset Value = $100 Firm Value = $100
             NOTE: Change in financial structure does not affect the amount
             or risk of the cash flows on the total package of debt and equity.
             WACC is 12.2% both before and after the change.
             • Change does affect the required rate of return on individual securities.

             • Risk is the same but shifted from one security holder to another.

             • Stockholders and bondholders required rates of return will change.

EB545 – Corporate Finance                                                            169
              Since firm carries less debt load, let's assume bondholder
              require less return - suppose it decreases from 8% to 7.3%.
                      rassets = debt rdebt + equity requity
                                  value        value
                      rassets = 30 (0.073) + 70 (requity ) = 0.122
                               100           100
                              So, requity = 14.3%

               NOTE: Lower leverage makes equity safer and reduces
               shareholder required rate of return.
               If we take this to the limit, where debt is equal to 0, then
                                  requity = 12.2%

EB545 – Corporate Finance                                                     170
    WACC and Taxes
                EXAMPLE: Tax rate (t) = 50%

                 Market Value    Market Yield   % of Cap. Struc.
    Debt         $400            0.08           40%
    Preferred    $100            0.10           10%
    Common       $500            0.12           50%
    Total        $1000                          100%

    After Tax Cost of Debt = 0.08 x (1-t) = 0.08 x 0.5 = 0.04
    (Because interest is deductible)

    WACC = (0.40 x 0.04) + (0.10 x 0.10) + (0.50 x 0.12) = 0.087

    Caution: WACC does not consider risk of project - while
    appropriately applied, CAPM does!

EB545 – Corporate Finance                                          171
    Financial Leverage - Effect on Beta
             • Stockholders and debtholders receive cash flow and share risk.
             • Debtholders bear much less risk than shareholders; for

               example, debt betas are near 0 for blue-chip stocks.
             • Firm's asset beta is equal to beta of firm's debt and equity

               (weighted average)
                   βassets = βportfolio = D/V(βdebt) + E/V(βequity)
             Consider Prior WACC Example where:
                   Asset Value = $100 Debt Value = $40
                                             Equity Value = $60
                   ---------------------     --------------------
                   Asset Value = $100 Firm Value = $100
                Assume that the βdebt = 0.2 and the βequity = 1.2
             then,      βassets = (0.4 x 0.2) + (0.6 x 1.2) = 0.80
             What happens after refinancing? Suppose βdebt falls to 0.1
               βassets = 0.80 = (0.3 x 0.1) + (0.7 x βequity)
             So, βequity = 1.1

EB545 – Corporate Finance                                                       172
    Asset Betas:
            Example: Morgan Company
            Beta = 2.0      rf = 0.10        E(rm) = 0.20
            Cost of Capital:
                 E(ri) = rf + βi(E(rm) - rf) = 0.30
            Suppose Year 1 payoff for asset i = $130
            Value of firm, PV0 = $130/1.30 = $100
            Now suppose the firm raises $100 to invest in the risk-free asset,
            which will have a payoff of $110:
                            PV0       C1         β      Req. ROR
          Co. Before Inv. $100        $130       2.0    0.30
          New Investment $100         $110       0      0.10
          Co. After Inv.    $200      $240       1.0    0.20
            NOTE: Beta for company after investment can be computed
            by weighted average.

EB545 – Corporate Finance                                                        173

                30%                   Company Befo

                            Company After

             r f (10%)

                            1.0      2.0

    Operating Leverage

EB545 – Corporate Finance                            174
             •   Fixed operating costs, so called because they accentuate
                 variations in profits.
             •   Just as commitment to fixed debt increases financial leverage,
                 commitment to fixed production charges adds to the beta of a
                 capital project.
             •   Cash flows generated by any productive asset are made up of
                 revenues, fixed costs and variable costs.
             •   Variable Costs: raw materials, sales commissions, some labor.
             •   Fixed costs: wages of workers under contract, property taxes.
             •   Fixed costs are cash outflows, whether asset is idle or not.
             •   Other things being equal, projects with higher ratio of fixed
                 costs to project value have higher project betas.
                 Operating Leverage =        % ∆EBIT
                                             % ∆Sales

         Selling Price per unit:       $20 (p)
         Variable Costs per Unit:      $12 (v)
         Fixed Operating Costs:        $60,000 (Fo)

EB545 – Corporate Finance                                                         175
           Interest Expense:            $25,000 (Ff)

           Degree of Operating Leverage at 15,000 units (x):
           DOL = % ∆ EBIT = x( p − v) =             15, 000(20 − 12)     = 2.0
                  % ∆ Sales     x( p − v) − FO 15, 000(20 − 12) − 60,000
               So, for each 1% change in sales( up or down), EBIT will
               change by two times that. Note that this is at a single point in

           Degree of Financial Leverage at 15,000 units (x):
                        DFL = % ∆Net Earnings
                                     % ∆EBIT
                     x( p − v) − FO        15, 000(20 − 12) − 60, 000
           DFL =                     =                                     = 1.71
                  x( p − v) − FO − Ff 15, 000(20 − 12) − 60, 000 − 25, 000
           •   For each 1% change in EBIT, net earnings will change by
               1.71 times that.
           •   The more fixed financial costs, the higher the degree of
               finanical leverage (DFL).

EB545 – Corporate Finance                                                           176
           Degree of Combined Leverage at 15,000 units:

           DCL =           x( p − v)        =     15, 000(20 − 12)      = 3.4
                    x( p − v) − ( Fo + F f ) 15, 000(20 − 12) − 85, 000
           DCL = DOL x DFL
               = 2.0 x 1.7
               = 3.4

           •   The more leverage you have, the riskier you are.
           •   Relationship between DOL and DFL is positively related.
           •   Higher DOL or DFL will generate a higher beta for the

EB545 – Corporate Finance                                                       177
           •   Financial leverage could include lease payments, preferred
               dividends, interest payments and some types of taxes.
           •   Different companies use different definitions for fixed item
               in DFL.
           •   Empirical tests confirm that companies with high operating
               leverage actually do have high betas.

EB545 – Corporate Finance                                                     178
Risk Analysis Techniques for Capital
    Sensitivity Analysis
             •   Pro forma cash flows are evaluated with specific variables in
                 the analysis forecasted at various levels, one at a time.
             •   For example, commodity price levels may be forecasted
                 under pessimistic, most likely and optimistic scenarios and
                 resulting NPV's calculated.
             •   Simple and useful method for ranking the relative impact on
                 project value from various variables.
             •   Useful in terms of establishing a floor value or worst case
             •   Major drawback is that only one variable's probability spread
                 can be evaluated at a time.
             •   NOTE: Underlying variables may be interrelated.

EB545 – Corporate Finance                                                        179
             •  Primary Objective is to define the distribution of profit (or
                NPV) which could be anticipated for the project.
             • Unlike sensitivity analysis, simulation considers the impact of

                changes on all variables simultaneously.
             • Accommodates for both independent and dependent random

      EXAMPLE: Consider an investment which ultimate net profit is a
        function of three separate, independent variables x, y & z. Suppose
        the relationship is defined as:
                        Pr ofit = 3.4 x − [( 1 + 17.9)(0.4 z 2 )]
                                             14 y
             where Profit is the dependent variable.

        Using an oil exploration example, x, y & z may be net pay thickness,
        recovery factors, productive area, drilling costs, crude price,
        operating expense, etc.
          Now suppose we know the exact values which x, y , z take on.

EB545 – Corporate Finance                                                        180
           Yes, this is a deterministic computation; no uncertainty with
             regard to the variables of interest.

           a. Given we live in an uncertain world, though, suppose all we
              know are ranges of outcomes for x, y, & z - which value do we
           b. Simulation allows us to model all the values in a range, for each
              random variable and compute a distribution of profit.

                   Pr ofit = 3.4 x − [( 1 + 17.9)(0.4 z 2 )]
                                        14 y

EB545 – Corporate Finance                                                         181
           f(x)                   f(y)

                      x                          y          z

                  RESULTS OF SIMULATION:

                     (-)     0             (+)

           NOTE: Consider full range and distribution of outcomes on
            each random variable - not just average or most likely

EB545 – Corporate Finance                                              182
           c. Actual state of nature: Variables x, y , z will have one specific
              value; howwever, before the decision or action we do not know
              what that value is.
           d. Do not use only mean or most likely case of each random
              variable to solve for "mean" or "most likely" profit; this
              approach ignores the uncertainty as defined by each distribution.
           e. There are a few special situations when the analytic approach
              (as opposed to simulation) is possible. For example, if we can
              find the PDF associated with each distribution, then insert into
              profit equation and solve.

EB545 – Corporate Finance                                                         183
           1. Define the distributions for each of the independent variables.
           2. Make a series of repetitive calculations for our profit expression.
           3. Each value is computed using a value of x, y & z selected from
              their respecitive distributions and represents one possible state
              of nature.
           4. Each repitition of the dependent variable calculation is called a
              "simulation pass".
           5. Values of the independent variables used for each pass are
              obtained through a random number sampling procedure which
              honors the shape and range of each variable's distribution.
           6. Passes are made until a sufficient number is made to define the
              profit distribution (i.e. at least 100, and as many as thousands.)

           NOTE: It is inappropriate to use same (single) random number to
            sample all distributions on a pass. Using same number implies
            fixed values for all variables.


EB545 – Corporate Finance                                                           184
           1. Capability to define uncertainty as a range and distribution of
              possible outcomes for each unknown factor.
           2. May be applied to any type of non-deterministic calculation.
           3. No limit to the number of variables that may be modeled.
           4. Distributions of random variables can be of any type.
           5. Different individuals can describe uncertainty about different
           6. Lends itself well to sensitivity analysis.
           7. Provides valuable inputs to decision tree modeling (by
              converting continuous to discrete distributios.)

    Independent versus Dependent Variables.

EB545 – Corporate Finance                                                       185
           EXAMPLE: Compare distributions of net pay thickness and initial
             production rates.
           ERROR: If the value of thickness sampled was high (which
             implies large reserves) and in the same pass, the IP sampled was
             very low then the resulting field life would be unrealistic,
           • Suppose that we are concerned about whether variables x and y

             are related to one another.
           Make a cross-plot of x versus y.

EB545 – Corporate Finance                                                       186
                      CASE OF INDEPENDENCE

           r.v. y

                                 r.v. x

           Sample each r.v. separately in the simulation pass.

EB545 – Corporate Finance                                        187
                        CASE OF COMPLETE DEPENDENCY

           r.v. y

                                                          y = f(x)

                                    r.v. x

           If we know x, then we know y (or vice versa); therefore, we
              only need specify one distribution in the simulation.

EB545 – Corporate Finance                                                188
                     CASE OF PARTIAL DEPENDENCY

           r.v. y

                               r.v. x
           1. x-y relationship not clearly independent or explicitly
           2. Variable y has a substantial range of values, given x - so we
              need to sample from y on each pass.
           3. Also, range of variation of ychanges as x increases.
           4. Therefore, the distribution of y changes as x changes and we
              must accommodate that change in the simulation.

EB545 – Corporate Finance                                                     189
         1. Prepare a cross-plot.
         2. Draw a boundary or envelope around the observed x-y data
            point. This boundary defines the limits within which x and y
            can vary.
         3. Any combination of x-y values outside the boundary have a
            zero probability of occurrence.
         4. Determine the variation of y within the boundary as a
            function of x and define a normalized distribution of y
            (0 to 1) which represents the observed variation.
         5. In this normalized distribution, 0 corresponds to the
            minimum value of y for a given value of x and 1.0
            corresponds to the maximum. Such that,
                                y − ymin
                       ynorm =
                               ymax − ymin
           where y = ymin, then 0 and y = ymax, then 1.0

           6. Inputs to the simulation model are:

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                 a. x distribution cumulative frequency;
                 b. cumulative frequency of the normalized y distribution;
                 c. value of y for each pass:

                  y = ymin + ( ymaxx − ymin x )( ynorm)

             yminx =   minimum value of y, given a sampled x
             ymaxx =   maximum value of y, given a sampled x
             ynorm =   value of the dimensionless, normalized y distribution
             y     =   computed value of r.v. y to use for each pass.

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Securities Markets
    •   Primary functions are resource allocation and
        exchange of assets.
    •   Role of Different Segments of Security Markets
    •   Mechanics of Securities Markets
             1.   Brokers and Brokerage Firms
             2.   Type of Accounts
             3.   Types of Orders

    •   Role of Regulation

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    Types of Security Markets
             1.   Primary or "New Issue" Market
                  •   Involves only new stock or bond issues.
                  •   Ex.: New sale of Texaco stock or Treasury bills.
                  •   Major component of primary market is the bond
                            About 3/4's of all new corporate issues are bonds.
                            ncluding government issues makes this percentage even higher.
                  •   Investment banker or underwriter
                            Middleman in the primary markets.
                            Ex.: Merrill Lynch or Salomon Bros.
                            For well-known issues, new issue is boughts by investment banker and then
                            resold to investors.
                  •   Underwriting a new issue
                            Gurantee of a successful issuance.
                            Underwriter syndicate utilized to share risk, purchase cost and geographical

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                  •   How is new issue price set?
                            Privately agreed upon
                            Auctioned competitively (i.e., state and local bond issues)
                            Prevailing prices for comparable issues
                            Current market conditions dictate an issue's price.

             2.   Secondary or "Used Issue" Market
                  •   Provides a means for trading existing securities among
                  •   Ex.: NYSE, OTC market, etc.
                  •   "Market-makers" - professionals who hold an
                      inventory of a stock.
                            Provides marketability
                            Allows for easy transfer from sellers to buyers.
                            "Specialists" on organized exchanges and "delaers" in the OTC.

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                 •    "Organized exchanges" provide a fixed location where
                      trading occurs.
                 •    "OTC" market is not physically located in one place;
                      it is a communications network.
                 •    Pricing Securities in an "Auction" market.
                            Prices determined by bidding of market participants called brokers.
                            Occurs in organized exchanges
                 •    Pricing securities in a "Negotiated" market
                            Involves a dealer network
                            Dealers stand ready to buy/sell securities at their specifed prices.
                            Dealers do have a vested interest because they buy (sell) securities from (to)
                            Dealer's profit comes from the difference in their "bid-ask spread" - the
                            price at which they will buy (bid) and sell (ask).
                            Ex.: OTC market for stocks and government securities.

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                 •    New York Stock Exchange (NYSE)
                            Largest and most prominent secondary market for stocks.
                            Over 200 issues are traded here.
                            Represents over 75% of the dolar volue traded in the U.S. ("Big Board")
                            1400 members - partners or directors of brokerage firms.
                            25% of members are specialists - assigned specific stocks and one of the 89
                            trading posts on the exchange's floor.
                 •    American Stock Exchange (AMEX)
                            Organization and trading procedures similar to NYSE.
                            Smaller nationally organized exchange.
                            Fewer seats (650) and stocks listed (1000)
                            Listing requirements are also less strict.
                            Smaller (especially energy and natural resource companies) dominate the
                            types of firms listed/
                 •    Regional Exchanges
                            Listing requirements are more lenient.
                            Specialize in smaller companies with limited geographical interest.
                            "Dual listings - chief attraction is lower commissions.
                            Ex.: Boston , Cincinnati, Midwest, Pacific Coast Stock Exchanges

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                 •    Over-the-Counter (OTC) Market
                            Smaller firms tend to be found in the OTC market (over 10,000)
                            Trades take place by telephone or other telecommunication devices.
                            NASD (National Association of Securities Dealers)
                            Markets exist for stocks, bonds, federal securities, municipal bonds,
                            negotiable CD's, mutual funds, etc.
                            Three levels: "national market", "regular market" and "additional OTC"
                            NASDAQ - The NASD Automated Quotation System - computerized
                            system that provides quotations on up to 3500 stocks.
                 •    Listed Options
                            Offer trading in standardized call and put option contracts.
                            Listed call-put options are traded on the Chicago Board Options Exchange
                            (CBOE), AMEX, NYSE, etc.

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             3.   Participating in the Market
                   •  Cash Account
                            Cash transactions only
                            "Type 1" account
                   •   Margin Account
                            Buy securities with a combination of cash and borrowed funds.
                            "Type 2" account
                            By using securities as collateral, you may borrow a certain percentage of the
                            purchase price, called margin.
                            Margin call - requirment to put up more collateral when doolar amount of
                            collateral falls below a certain level.
                            Note: Using margins can increase or decrease your potential investment
                            returns - "leveraged" investment.
                            Because margin magnifies both gains and losses, using a margin account is
                            a much riskier type of investing.
                                  Ex.: Assume:
                                  1. You buy 200 Exxon shares at $50 per share.
                                  2. Margin requirement = 50%
                                  3. Broker loan rate = 8%

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                   •   Asset Management Account
                            Combines checking, credit card, money market and margin accounts.
                            Funds are moved among accounts to minimize your interest expense and
                            maximize short-term yields.
                            Simplifies cash management problems.
                   •   "Street-Name" Account
                            Securities are left with a broker
                            Offers safe storage and makes trading easier.
                            Alternative to taking delivery.

               3. Types of Trade Positions

                   •   Long Position
                            Involves initially purchasing a position in the hope that the security's price
                            will increase.
                            Objective in being long is to buy low and sell high.
                            Entitles you to any dividends or interest paid plus the capital gains.
                            Strategy when you are "bullish" about a security's future prospects.

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                   •   Short Position
                            Involves initially selling a security in the hope that the security will decline
                            in price.
                            Objective is still to buy low and sell high - however, you reverse the order
                            of the transaction.
                            "Bearish" strategy.
                                  1.    Your broker borrows the security or shares from the account of
                                  another customer (or brokerage firm).
                                  2.     He then sells the security to you.
                                  3.     Sale proceeds are held in escrow until you liquidate your
                                  position by buying back the (replacing the borrowed) security.
                                  4.    If the security is purchased at a lower price in the future, then
                                  you realize a capital gain and vice versa.

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             5.   Types of Orders
                   •  Market Order - Buy or sell ASAP at the best price
                   •  Limit Order - Restricts the order to a specified price.
                   •  Stop-Loss (Stop) Order - An order to buy (sell) a
                      security when its market price trades at or above
                      (below) a specified level.

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    Regulation of Securities Markets
             Securities Act of 1933
                 •    Major Objective - Full disclosure in order to protect
                      investors from fraud and to provide them with a basis
                      for informed judgments.
                 •    Applies to all interstate offerings to the public over a
                      certain amount ($1.5 million). Note: Does not apply to
                      government and regulated industry offerings.
                 •    Securities must be registered at least 20 days before
                 •    Company must issue a prospectus.

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             Securities Act of 1934
                 •    Major Objective - Applies full disclosure principle to
                      secondary securities markets.
                 •    Creation of the Securities and Exchange Commission
                 •    SEC's purpose is to regulate the organized securities
                 •    Requires registration and regulation of national
                      security exchanges.
                 •    Regulates insider trading.
                 •    Watches for price manipulation.
                 •    Gives SEC power over proxy machinery and
                 •    Gives Federal Reserve Board the power to set margin

             Blue-Sky Laws

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                 •    State laws which govern securities issues and matters
                      in individual states.
                 •    Company must be registered or "blue-skyed" in state
                      to issue shares there.
                 •    Individual states have tougher rules and regulations on
                      the sale of securities than does the SEC.

             Industry Self-Regulation
                 •    National Association of Security Dealers (NASD)
                 •    Individual exchanges
                 •    Control trading practices among their members.

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Financing vs. Investment Decisions
    •   Financing decisions involve the firm directly in
        capital markets.
    •   Capital Markets: Any activity which serves to
        allocate resources to their most productive
    •   Capital markets generally work well with
        respect to the efficient allocation of resources.

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Differences between Investment and
Financing Decisions
    1. Not the same degree of finality; financing
       decisions are easier to reverse, i.e. higher
       abandonment value.
    2. More difficult to generate rents or positive
       NPV's with financing decisions - more
       competitive market.
             •   All firms compete in financial market.
             •   Investment market is differentiated in terms of competition.

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Efficient Capital Markets
    •   Def.: An efficient capital market is one in which
        market prices fully reflect all information
        available at that time ("well functioning").
    •   If capital markets are efficient, then purchase or
        sale of any security at the prevailing market
        price is never a positive NPV transaction.
    •   Implication is that information is widely and
        cheaply available to investors and that all
        relevant and ascertainable information is
        already reflected in security prices.

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    1. Abnormal Returns - An abnormal return of firm
       i at time t is the deviation between its realized
       return and the expected return as per CAPM
    2. "Beating the Market" (risk-adjusted basis) -
       Must take risk into consideration before
       deciding if you "beat the market".
    3. Ex ante vs. ex post market efficiency - The key
       here is that if the information had been available
       to you ex ante could you have earned abnormal
    4. Consistency - Are abnormal returns luck or
       consistent behavior?

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Three Forms of Market Efficiency
    1. Weak Form - Def.: Prices reflect all information
       contained in the record of past prices.
        •    "Random-walk" model
             a.   Prices changes are random.
             b.   Given a price, the size or direction of next change is

        •    "Technical Rules"
             a.   Used by investors to exploit "patterns" they claim to see in
                  stock prices.
             b.   Weak form would suggest these are inappropriate rules
                  because there is no useful information in the sequence of
                  past changes in stock prices.

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        •    Tests of Random-Walk Model
             a.   Is there statistical evidence that the "random-walk" model
                  does not hold?
             b.   Is there economic significance?
                  •    Serial correlation studies: Is today's price significantly related to yesterday's price?
                  •    Result: There is slight statistical significance but not economic significance.
                  •    Runs Test: Are there more positive or negative price changes over a given period
                       than at random? (Looking at a string of price changes.)
                  •    Results: No difference than at random.

        •    Implications of Random Walk or Weak Form
             a.   If prices always reflect all relevant information, then they
                  will change only when new information arrives.
             b.   However, new information by definition cannot be
                  predicted ahead of time.
             c.   If stock prices already reflect all that is predictable, then
                  stock price changes must reflect only the unpredictable.
             d.   Therefore, the series of changes must be random.

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        •    Two Important Notes Concerning Random
             Walk Model:
             1.   Randomness is often confused with irrationality;
                  however, randomness is the predicated result of
                  rational valuation.
             2.   Random Walk Hypothesis does not say actual
                  prices are random but that price changes are
                  random (size and direction).

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    2. Semi-Strong Form - Def.: Prices reflect not only
       past prices, but all other published or publicly
       available information.
        •    Empirical Tests:
             a.   Reaction to earnings announcements.
             b.   Reaction to stock splits and dividend announcments.
             c.   Effects of secondary distributions
                  Large blocks sold outside of market hours generates
                  "combination effect"
                      Liquidity effect drives price down because it is a large block (temporary
                      Information effect drives price down but does not come up.
                      Combination effect is that price drops and comes back partially,

        NOTE: Key here is that most of this information was rapidly and
        accurately impounded in the price of the stock.

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        •    Anomalies (Risk-Adjusted Basis):
             a.   P/E Effect - Low P/E stocks have positive abnormal
                  returns; high P/E stocks have negative abnormal returns.
             b.   Size Effect - Small firms on average have earned positive
                  abnormal returns; large firms, on average, earn negative
                  abnormal returns.
                  NOTE: Both a & b could be caused by the fact that there is more information risk in
                  small firms.
             c.   January effect - Firms tend to do better in January
                  (generate higher returns).
             d.   Day of the Week Effect - Monday's tend to have lower
                  returns than other days of the week.
             e.   Value Line Investment Survey - Abnormal returns could be
                  earned if you could trade the instant Value-Line was
        NOTE: Semi-strong form is most commonly accepted
        form by academics and practitioners.

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    3. Strong Form - Def.: Prices reflect all information
       contained in the record of past prices.

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